**2. Problem Formulation**

Contemplate an incompressible two-dimensional stagnation-point flow of a Sutterby nanofluid across a stretching/shrinking sheet as shown in Figure 1, where *x* and *y* are the Cartesian coordinates with the *x*-axis positioned in the horizontal direction, and the *y*-coordinate is normal to the *x*-coordinate. The free stream velocity is denoted by *ue*, and *uw* signifies the velocity of the moving sheet, where *uw* > 0 infers the state of the stretching sheet, *uw* < 0 the embodies shrinking sheet, and *uw* = 0 typifies the

stationary sheet. The moving (stretching or shrinking) sheet is penetrable and there is a uniform surface mass flux, of velocity *vw*, with *vw* > 0 to imply the injection situation and *vw* < 0 for the suction state. The free stream temperature and the wall temperature are denoted by *T*∞ and *Tw*, respectively.

**Figure 1.** Schematic diagram of the present problem: (**a**) shrinking sheet (*uw* < <sup>0</sup>); (**b**) stretching sheet (*uw* > <sup>0</sup>).

Sutterby [1,2] introduced the constitutive law for the Sutterby fluid by expressing the Cauchy stress tensor (**T**) as:

$$\mathbf{T} = -p\mathbf{I} + \mathbf{S},\tag{1}$$

where *p* is the pressure, **I** is the identity vector, and **S** is the extra stress tensor which can be defined as follows [21]:

$$\mathbf{S} = \mu\_0 \left[ \frac{\sinh^{-1} \left( E \dot{\boldsymbol{\gamma}} \right)}{\left( E \dot{\boldsymbol{\gamma}} \right)} \right]^m \mathbf{A}\_1. \tag{2}$$

Here, μ0 is the viscosity at low shear rates, *E* is the material time constant, .γ = +tr(**<sup>A</sup>**1)2/2 is the second invariant strain tensor, **A**1 is the first order Rivlin–Erickson tensor or deformation rate tensor which is defined as **A**1 = (∇**V**) + (∇**V**)T, and *m* is the power-law index. The Sutterby model in Equation (2) is a versatile model when the value of *m* changes. For instance, when *m* = 0, the Sutterby model imitates the Newtonian fluid behavior, when *m* = 1, the model is reduced to the Eyring model, and this model also predicts specifically the pseudo-plastic (shear thinning) and dilatant (shear thickening) fluid properties when *m* > 0 and *m* < 0, respectively. By taking the velocity field as **V** = [*u*(*<sup>x</sup>*, *y*), *<sup>v</sup>*(*<sup>x</sup>*, *y*)], and under the assumptions mentioned earlier, the governing boundary layer equations in the dimensional form can be formed as follows [36]:

$$\frac{\partial \overline{u}}{\partial \overline{x}} + \frac{\partial \overline{v}}{\partial \overline{y}} = 0,\tag{3}$$

$$
\overline{u}\frac{\partial\overline{u}}{\partial\overline{x}} + \overline{v}\frac{\partial\overline{u}}{\partial\overline{y}} = \frac{\mu\_{nf}}{\rho\_{nf}}\frac{\partial^2\overline{u}}{\partial\overline{y}^2} + \frac{\mu\_{nf}}{\rho\_{nf}}\frac{mE^2}{2}\left(\frac{\partial\overline{u}}{\partial\overline{y}}\right)^2\frac{\partial^2\overline{u}}{\partial\overline{y}^2} + \overline{u}\_\varepsilon\frac{d\overline{u}\_\varepsilon}{d\overline{x}} - \frac{\sigma B\_0^2(\overline{u}\_\varepsilon - \overline{u})}{\rho\_{nf}},\tag{4}
$$

$$
\overline{u}\frac{\partial T}{\partial \overline{\mathbf{x}}} + \overline{v}\frac{\partial T}{\partial \overline{y}} = \frac{k\_{\text{nf}}}{\left(\rho \mathbb{C}\_{p}\right)\_{\text{nf}}} \frac{\partial^{2} T}{\partial \overline{y}^{2}} + \frac{16}{3} \frac{\sigma\_{1} T\_{\text{os}}^{3}}{\left(\rho \mathbb{C}\_{p}\right)\_{\text{nf}} k\_{1}} \frac{\partial^{2} T}{\partial \overline{y}^{2}},\tag{5}
$$

along with the respective boundary conditions:

$$\begin{array}{llll}\overline{u} = \overline{u}\_{\overline{w}\prime} & \overline{v} = \overline{v}\_{\overline{w}\prime} & T = T\_{\overline{w}\prime}(x) & \text{at} & \overline{y} = 0. \\\overline{u} = \overline{u}\_{t\prime} & \frac{\partial \overline{u}}{\partial \overline{y}} \to 0, & T \to T\_{\infty} & \text{as} & \overline{y} \to \infty, \end{array} \tag{6}$$

where *u* and *v* denote velocity components in the *x* and *y* directions, respectively, μ*n f* is the dynamic viscosity of the nanofluid, σ is the electrical conductivity, *B*0 is the magnetic field strength, ρ*n f* is the density of the nanofluid, σ1 is the Stefan Boltzmann constant, *k*1 is the Rosseland mean absorption coefficient, *kn f* is the thermal conductivity of the nanofluid, and -*Cpn f* is the specific heat capacity of the nanofluid. The detailed definitions of the nanofluid parameters are given by the following expressions, which are valid when the nanoparticles are of spherical shape or similar to a spherical shape [37]:

$$\begin{aligned} \mu\_{nf} &= \frac{\mu\_{bf}}{\left(1-\phi\right)^{2.5}}, \quad \alpha\_{nf} = \frac{k\_{nf}}{\left(\rho \mathbb{C}\_{p}\right)\_{nf}}, \quad \left(\rho \mathbb{C}\_{p}\right)\_{nf} = \left(1-\phi\right)\left(\rho \mathbb{C}\_{p}\right)\_{bf} + \phi\left(\rho \mathbb{C}\_{p}\right)\_{s}, \\\ \rho\_{nf} &= \left(1-\phi\right)\rho\_{bf} + \phi\left.\rho\_{sf}\right. \quad \frac{k\_{nf}}{k\_{bf}} = \frac{\left(k\_{\*}+2\left.k\_{bf}\right)-2\left.\phi\left(\left.k\_{bf}-k\_{\*}\right)\right.\right.}{\left(k\_{\*}+2\left.k\_{f}\right)+\phi\left(\left.k\_{f}-k\_{\*}\right)}\right.\end{aligned} \tag{7}$$

where φ denotes the nanoparticle volume fraction, μ*b f* denotes the dynamic viscosity of the base fluid, α*n f* denotes the thermal diffusivity of the nanofluid, *kb f* is the thermal conductivity of the base fluid, *ks* is the thermal conductivity of the solid fractions, *Cp* is the specific heat capacity, and ρ*b f* and ρ*s* are the densities of the base fluid and solid fractions, respectively. The Sutterby model reflects the dilute polymer solution where the polymer is diluted in the appropriate solvent. Hence, for the present study, *n*-Hexane is chosen as the base fluid (solvent). Table 1 displays the specific values for the respective thermophysical features of *n*-Hexane and magnetite nanofluid [38].

**Table 1.** The thermo physical characteristics of the essential fluid and nanoparticles.


### **3. Non-Dimensionalization of the Governing Equations**

Considering the following the non-dimensional variables:

$$\begin{array}{ll}\mathbf{x} = \frac{\mathbf{u}\overline{\mathbf{u}}}{\mathbf{u}\_{0}}, & \mathbf{y} = \sqrt{\frac{\mathbf{u}}{\mathbf{v}\_{bf}}}\overline{\mathbf{y}}, & \mathbf{u} = \frac{\overline{\mathbf{u}}}{\mathbf{u}\_{0}}, & \mathbf{u}\_{\overline{\mathbf{u}}} = \frac{\overline{\mathbf{u}}\_{\mathbf{u}}}{\mathbf{u}\_{0}}, & \mathbf{v} = \frac{\overline{\mathbf{v}}}{\sqrt{\mathbf{u}\mathbf{v}\_{bf}}},\\\mathbf{v}\_{\overline{\mathbf{w}}} = \frac{\overline{\mathbf{v}}\_{\mathbf{w}}}{\sqrt{\mathbf{u}\mathbf{v}\_{bf}}}, & \mathbf{u}\_{\mathbf{f}} = \frac{\overline{\mathbf{u}}\_{\mathbf{f}}}{\mathbf{u}\_{0}}, & \boldsymbol{\Theta} = \frac{\overline{\mathbf{T}} - \mathbf{T}\_{\mathbf{w}}}{\mathbf{T}\_{0}},\end{array} \tag{8}$$

where *u*0 is the characteristic velocity and introducing the stream function ψ, which can be defined by *u* = ∂ψ∂*y* and *v* = <sup>−</sup>∂ψ∂*x* , Equations (4) and (5) become:

$$\frac{\partial\psi}{\partial y}\frac{\partial^2\psi}{\partial x\partial y} - \frac{\partial\psi}{\partial x}\frac{\partial^2\psi}{\partial y^2} = \frac{A\_1}{A\_2}\frac{\partial^3\psi}{\partial y^3} + \frac{A\_1}{A\_2}\frac{m\text{De}}{2}\left(\frac{\partial^2\psi}{\partial y^2}\right)^2\frac{\partial^3\psi}{\partial y^3} + \mu\_\varepsilon\frac{d\mu\_\varepsilon}{dx} - \frac{\sigma B\_\vartheta^2}{\rho\_{bf}a}\frac{1}{A\_2}\left(\mu\_\varepsilon - \frac{\partial\psi}{\partial y}\right) \tag{9}$$

$$\frac{\partial\psi}{\partial y}\frac{\partial\theta}{\partial x} - \frac{\partial\psi}{\partial x}\frac{\partial\theta}{\partial y} = \frac{A\_4}{A\_3}\frac{1}{Pr}\frac{\partial^2\theta}{\partial y^2} + \frac{1}{A\_3}\frac{4}{3}\frac{Rd}{Pr}\frac{\partial^2\theta}{\partial y^2},\tag{10}$$

with the corresponding boundary conditions:

$$\begin{array}{llll}\frac{\partial\psi}{\partial y} = \mu\_{\text{W}} & \frac{\partial\psi}{\partial x} = -\upsilon\_{\text{w}\_{\text{V}}} & \theta \, T\_{0} = T\_{\text{w}}(\text{x}) & \text{at} & y = 0, \\\frac{\partial\psi}{\partial y} \to u\_{\text{\'e}} & \frac{\partial^{2}\psi}{\partial y^{2}} \to 0, & \theta \to 0 & \text{as} & y \to \infty, \end{array} \tag{11}$$

while satisfying the continuity equation of Equation (3). In Equations (9) and (11), *M* = σ*<sup>B</sup>*20 ρ*b f a* is the magnetic parameter, *Rd* = <sup>4</sup>σ1*T*3∞ *k*1*kb f* is the radiation parameter, Pr = μ*b <sup>f</sup>*(*cp*)*b f kb f* is the Prandtl number, *De* = *u*20*aE*<sup>2</sup> <sup>ν</sup>*b f* is the Deborah number, φ is the nanoparticle volume fraction, and terms *A*1, *A*2, *A*3, and *A*4 are expressed as:

$$\begin{split} A\_{1} &= \frac{1}{\left(1-\phi\right)^{2.5}}, \quad A\_{2} = 1-\phi+\phi\frac{\rho\_{s}}{\rho\_{bf}}, \quad A\_{3} = 1-\phi+\phi\frac{\left(\rho c\_{p}\right)\_{s}}{\left(\rho c\_{p}\right)\_{bf}},\\ A\_{4} &= \frac{k\_{nf}}{k\_{f}} = \frac{k\_{\*}+2k\_{bf}-2\phi\left(k\_{bf}-k\_{\*}\right)}{k\_{bf}+2k\_{bf}+\phi\left(k\_{bf}-k\_{\*}\right)}.\end{split} \tag{12}$$

The functions *uw*, *vw* and *Tw*(*x*) are assumed to be in the following form to ensure that similarity solution exists:

$$u\_{\overline{w}} = \frac{\mathcal{U}\_1}{\mathcal{U}\_0} \mathbf{x}^{\frac{2}{3}}, \quad v\_{\overline{w}} = \frac{\mathcal{U}\_1}{\sqrt{\mathcal{U}\mathcal{V}\_{bf}}} \mathbf{x}^{-\frac{2}{3}}, \quad T\_w(\mathbf{x}) = T\_0 \mathbf{x}^{\frac{2}{3}}, \tag{13}$$

where *u*1 is the reference velocity, *v*1 is the normal reference velocity, and *T*0 is the reference temperature.
